Calder\'on-Zygmund operators on Zygmund spaces on domains

Given a bounded Lipschitz domain $D\subset \mathbb{R}^d$ and a Calder\'on-Zygmund operator $T$, we study the relations between smoothness properties of $\partial D$ and the boundedness of $T$ on the Zydmund space $\mathcal{C}_{\omega}(D)$ defined for a general growth function $\omega$. In the proof we obtain a T(P) theorem for the Zygmund spaces, when one checks boundedness not only of the characteristic function, but a finite collection of polynomials restricted to the domain. Also, a new form of extra cancellation property of the even Calder\'on-Zygmund operators in polynomial domains is stated.Comment: 31 page

Irregular nonlinear operator equations: Tikhonov's regularization and iterative approximation

A problem of iterative approximation is investigated for a nonlinear operator equation regularized by the Tikhonov method. The Levenberg-Marquardt method, its modified analogue, and the steepest descent method are used. For the first and second methods the regularizing properties of iterations are established and the error of approximate solution is given. For the third method it was proved that iterations are stabilized in a neighborhood of the required solution and satisfy the strong Fejйr property. © 2013 by Walter de Gruyter Berlin Boston 2013

The Levenberg-Marquardt method and its modified versions for solving nonlinear equations with application to the inverse gravimetry problem

The Levenberg-Marquardt method and its modified versions are studied. Under some local conditions on the operator (in a neighborhood of a solution), strong and weak convergence of iterations is established with the solution error monotonically decreasing. The conditions are shown to be true for one class of nonlinear integral equations, in particular, for the structural gravimetry problem. Results of model numerical experiments for the inverse nonlinear gravimetry problem are presented. © 2013 Pleiades Publishing, Ltd